# Cointegration and Adjustment in the CVAR(∞) Representation of Some Partially Observed CVAR(1) Models

## Abstract

**:**

## 1. Introduction

## 2. The Assumptions and Main Results

**Assumption**

**1.**

#### The Main Results

**Theorem**

**1.**

**Proof**

**Theorem**

**of**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

## 3. Two Examples of Simplifying Assumptions

**Case**

**1**

_{12}= 0)

**.**If the unobserved process ${X}_{2t}$ does not cause the observation ${X}_{1t},$ then ${M}_{12}=0.$ Therefore, ${M}_{12}{V}_{2T}=0$ and from (20) it follows that

**Definition**

**1.**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

**Case**

**2**

_{2}= 0, and M

_{12}and C

_{1}are strongly orthogonal)

**.**Because ${C}_{2}=0$ and ${M}_{21}^{\prime}{\mathsf{\Omega}}_{1}^{-1}{C}_{1}=0,$ Lemma 1 shows that ${V}_{2T}=0,$ so that the condition ${M}_{12}{V}_{2T}=0$ and (20) hold. Moreover, strong orthogonality also implies that ${M}_{12}^{\prime}{C}_{1}=0$ such that ${M}_{12}={C}_{1\perp}\xi $ for some $\xi .$ Hence

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A.

**Theorem**

**A1.**

**Proof**

**of**

**Theorem**

**A1.**

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**MDPI and ACS Style**

Johansen, S.
Cointegration and Adjustment in the CVAR(∞) Representation of Some Partially Observed CVAR(1) Models. *Econometrics* **2019**, *7*, 2.
https://doi.org/10.3390/econometrics7010002

**AMA Style**

Johansen S.
Cointegration and Adjustment in the CVAR(∞) Representation of Some Partially Observed CVAR(1) Models. *Econometrics*. 2019; 7(1):2.
https://doi.org/10.3390/econometrics7010002

**Chicago/Turabian Style**

Johansen, Søren.
2019. "Cointegration and Adjustment in the CVAR(∞) Representation of Some Partially Observed CVAR(1) Models" *Econometrics* 7, no. 1: 2.
https://doi.org/10.3390/econometrics7010002